Stability-Regularized Nested CV

• Stability-Regularized Nested CV is a framework that integrates stability penalties into nested cross-validation to improve model selection and risk estimation.
• It combines traditional CV error minimization with empirical measures of algorithmic stability to address the adaptivity gap between validation and test performance.
• Empirical studies indicate that SR-nCV enhances test-MSE and support recovery for unstable models, providing finite-sample guarantees in high-dimensional contexts.

Stability-Regularized Nested Cross-Validation (SR-nCV) defines a framework for model selection and risk estimation that explicitly incorporates algorithmic stability into the nested cross-validation procedure. By blending traditional cross-validation error minimization with empirical stability penalties, SR-nCV addresses the well-documented adaptivity gap: the risk that strong validation-set performance occurs alongside unpredictable out-of-sample behavior due to model instability. This approach operationalizes theoretical advances in CV stability and provides finite-sample guarantees for model selection and prediction error control in modern, potentially high-dimensional contexts (Cory-Wright et al., 11 May 2025, Lei, 29 May 2025, Fang et al., 2013).

1. Algorithmic Stability: Definitions and Measures

Stability in statistical learning quantifies the sensitivity of a model’s predictions or risk to perturbations in the training dataset. Two central notions appear in the literature:

• Leave-One-Out (LOO) Lq-Stability: For any real-valued functional $h$ on datasets, the LOO L$_q$-stability is $$S^{loo}_{q,n}(h) = \|h(D_n) - h(D_n^{(-n)})\|_{L_q}$$, where $D_n^{(-n)}$ omits one data point. This measures the $L_q$-norm of the discrepancy induced by removing a single observation.
• Perturb-One (PO) Lq-Stability: $S^{po}_{q,n}(h) = \|h(D_n) - h(D_n^{(i)})\|_{L_q}$ replaces one point with an i.i.d. copy, thus reflecting model robustness to data perturbation rather than removal.

Empirical approximations of these quantities underpin the stability penalties in SR-nCV (Lei, 29 May 2025). In practice, for supervised learners $\beta(\theta)$, stability is typically measured as the maximum—or average—change in prediction or loss across all folds when a fold is omitted from the training set:

$$\hat{\mu}_h(\theta) = \max_{j=1,\ldots,k} \frac{1}{n} \sum_{i=1}^n \left| \ell(y_i,\beta^{(N_j)}(\theta;x_i)) - \ell(y_i,\beta(\theta;x_i)) \right|$$

where $\beta^{(N_j)}(\theta)$ is retrained without fold $N_j$ (Cory-Wright et al., 11 May 2025).

2. Stability-Regularized Objective Functions in Nested CV

Stability-regularized nested cross-validation replaces the standard inner validation criterion with a convex combination of the empirical validation risk and a stability measure. For hyperparameter $\theta$ and regularization weight $\lambda \geq 0$, the objective within the inner loop is:

$$\theta^*(\lambda) \in \arg\min_{\theta \in \Theta} \left\{ h(\theta) + \lambda\, \hat{\mu}_h(\theta) \right\}$$

where $h(\theta)$ is the average k-fold cross-validation error, and $\hat{\mu}_h(\theta)$ is the empirical model-stability measure (Cory-Wright et al., 11 May 2025). The weight $\lambda$ controls the tradeoff between predictive accuracy and robustness to resampling.

In the classical "PASS" setting for sparse regression and variable selection, an alternative criterion combines Cohen's Kappa (agreement of support sets under resampling) and cross-validation error as a ratio, with the tuning parameter chosen by maximizing this PASS score (Fang et al., 2013).

3. Nested CV Implementation with Stability Regularization

The SR-nCV paradigm generally unfolds as follows:

1. Outer Loop: Partition data into $K_{out}$ outer folds. For each fold, reserve one subset as the test set.
2. Inner Loop: On the training split, carry out an inner $K_{in}$-fold cross-validation. For each candidate $\theta$ and each penalty $\lambda$ in a grid:
   • Compute both the inner cross-validation error and empirical stability.
   • Score each $\theta$ according to the stability-regularized criterion.
   • Select the optimal $\theta_{\text{inner}}^*$ for each $(\lambda,\, \text{fold})$.
3. Model Selection: For each $\lambda$, average the outer fold validation errors to obtain $S(\lambda)$. Select $\lambda^* = \arg\min_\lambda S(\lambda)$.
4. Final Estimation: With $\lambda^*$ fixed, perform a full-data stability-regularized CV to select $\theta^*$. Retrain on all data if desired.

This formalism admits a variant for support recovery in sparse regression, where the PASS criterion is optimized in each outer fold and final models are re-fitted on all samples (Fang et al., 2013).

The following high-level pseudocode synthesizes the procedure (Cory-Wright et al., 11 May 2025, Lei, 29 May 2025):

Step	Description	Key Quantities
1	Outer split	Outer folds $N_1,\ldots,N_k$
2	Inner SR-CV	$h(\theta),\,\hat\mu_h(\theta)$
3	Model selection	$\lambda^*,\,\theta^*$
4	Retraining	Final model on all data

Implementation tips include warm-starting along $\theta$ and parallelizing over folds or grid values to mitigate computational cost (Cory-Wright et al., 11 May 2025, Fang et al., 2013).

4. Theoretical Guarantees and Consistency

Theoretical justification for SR-nCV is grounded in stability-based risk concentration theorems. Specifically, under uniform Lq-stability, the following properties hold (Lei, 29 May 2025):

• Consistency: If $\gamma=O(1)$ and the stability penalty vanishes uniformly over $\theta$, then the minimizer of the penalized criterion converges in probability to the oracle risk minimizer. For proper tuning sequences, selectors are asymptotically consistent even when stability regularization is present.
• Risk Bounds: For bounded loss functions ($\ell \leq M$) and true stability $\mu_h$, for any train/test split, with probability at least $1-\Omega$ the test error is controlled by

$$\mathrm{TestError}(\beta(\theta)) \leq \frac{1}{n} \sum_{j=1}^k h_j(\theta) + \sqrt{ \frac{M^2 + 6M k\,\mu_h}{2 k \Omega} }$$

demonstrating that minimizing both validation error and empirical stability improves generalization error (Cory-Wright et al., 11 May 2025).

• Model Selection Consistency (PASS): Under regularity conditions for LASSO, SCAD, and adaptive LASSO, maximizing the PASS score in the defined asymptotic regime leads, with probability tending to one, to selection of the support coinciding with the true support set (Fang et al., 2013).

5. Empirical Performance and Application Domains

Empirical results demonstrate the utility of SR-nCV and related variants. On UCI-style regression datasets:

• In sparse ridge regression, SR-nCV improved average relative test-MSE by 4% versus standard k-fold CV, with improvements reaching 10% in overdetermined and 4.85% net across all regimes.
• For CART, improvements averaged 4.1% test-MSE overall.
• In both settings, SR-nCV reduced the CV–test adaptivity gap, i.e., the underestimation of test error by CV was reduced dramatically.
• No meaningful improvements were observed for XGBoost, supporting the premise that SR-nCV is most valuable for unstable or interpretable models (Cory-Wright et al., 11 May 2025).

PASS demonstrated improved support recovery and outperformed standard model selection tools such as BIC, Cp, 10-fold CV, and GCV, particularly in moderate signal-to-noise settings and scenarios with $p$ fixed or $p \approx \sqrt{n}$ (Fang et al., 2013).

6. Methodological Variants and Implementation Guidance

Multiple specific protocols for stability-regularized nested CV exist:

• PASS: Combines Cohen's Kappa for model support stability with cross-validation error in a ratio. It explicitly penalizes degenerate selections (all-zero or full support) and uses repeated random data splits.
• Weighted Additive Penalties: Employ a scalar penalty $\lambda$ (or $\gamma$) on empirical stability, selected via grid search or nested tuning.
• Stability Measurement: Empirical L$_q$ norms, "perturb-one" (replace-a-point) evaluations, or subset averaging are standard. For some M-estimators, stability can be estimated via gradients to reduce computation (Lei, 29 May 2025).
• Computational Cost: Complexity is roughly $$|\Lambda| \times K_{out} \times |\Theta| \times K_{in}$$ model fits plus stability replicates. Parallelization and warm starting are recommended. Excessive penalty or over-refined hyperparameter grids should be avoided.

Crucial implementation considerations include random split management for reproducibility, protection against degenerate models, and post-selection inference for uncertainty quantification (Fang et al., 2013, Cory-Wright et al., 11 May 2025).

7. Limitations and Ongoing Directions

Limitations of SR-nCV are context dependent:

• In severely underdetermined regimes (e.g., very high $p/n$), additional regularization may marginally deteriorate performance if models are already highly stable.
• Additional computational overhead arises from the outer grid over penalty weights, but this is mitigated by use of coarse grids and parallel methods.
• Empirical effects for black-box or intrinsically stable models (such as XGBoost) are negligible, confirming that SR-nCV's benefit is model-context specific (Cory-Wright et al., 11 May 2025).

Empirical studies of stability-regularized nested CV for non–linear and high-dimensional settings, as discussed in contemporary theorization (Lei, 29 May 2025), remain an open direction. Comparative studies with ordinary nested CV and stability-tuned alternatives are needed to delineate optimal use cases, penalty calibration strategies, and the potential for further improvements in risk/variance characterization.

• "Stability Regularized Cross-Validation" (Cory-Wright et al., 11 May 2025)
• "A Modern Theory of Cross-Validation through the Lens of Stability" (Lei, 29 May 2025)
• "A note on selection stability: combining stability and prediction" (Fang et al., 2013)